In the previous chapters we have seen how the Laws of Thought, as one of
the basic tenets of Logic, are in fact quite troublesome and not at all
``obvious truths''. Since they are generally used to determine
truth in an extensible manner, this is a problem. We also saw that to
make the way they are usually framed in English (at least) precise
and to not admit existential paradoxes, we had to try to formulate
them in terms of existential set theory (where set theory in
general is ``the mother of all reasoning systems''). Only in
existential set theory (with a well-defined set Universe) could the
abstractions of ``being'' or ``non-being'' be found, only by extending this set theory with the null Not a Set (while leaving
the empty set as an element of set theory within the set-theoretic
Universe in question) could we deal with certain paradoxical statements
that appear to be perfectly understandable, well-formed English
sentences that parse out logically to be nonsense, contradiction,
to things that violate our intuitive ideas of true and false, existence
or non-existence, or that simply fail to actually specify any set in our
set Universe including the empty set.

In this chapter we'll home in on Logic per se. The purpose of this book
is in no sense to denigrate the power, the beauty, the simplicity of
Logic (or its cousin Mathematics); it is to point out that it is a
sterile kind of beauty that cannot in and of itself give rise to
a single absolute truth relevant to the physical world we live in.
There is a fundamental disconnect between experience and reason that
cannot be filled in by reason.

The easiest way to accomplish our goals will be by presenting a very few
examples of logical arguments (famous logical arguments at that) to
illustrate the different parts of a system of logic. We will see
that systems of logical inference, without exception, can be described
in terms of actions taken on sets (no surprise), that rules of inference
are in some sense set-theoretical definitions or operations, and that to
go beyond the elementary list-making and categorization operations
of a raw set theory we have to dress the sets up with a mix of
definitions and axioms.

A language is often viewed as a system of definitions, a dictionary.
All dictionaries4.1 share a fundamental self-referentiality that makes
them far from trivial logical objects. Tarzan aside, it isn't at all
obvious that real human beings are capable of taking a dictionary
for a strange language alone and learning the language thus
represented4.2. Indeed, there is considerable evidence to the contrary -
without a Rosetta stone, without a context or pre-existing relationship
in terms of which a decryption algorithm can seek information
compressing patterns, the dictionary is arbitrary and can even
continuously change. Modern cryptography is based on this premise
- it constructs a highly nontrivial ``dictionary'' so that statements
are (ideally) indistinguishable from random noise, at which point there
is no informational compression at all. A further problem is that even
within a single language with a fixed dictionary, all dictionary
definitions in that language are circular - they are written in
words in the dictionary, whose definitions are written in terms of other
words, which are written in terms of other words, until eventually
you find that the dictionary is nothing but a set of equivalence
connections with a certain pattern. Yessir, Tarzan's accomplishment
puts John von Neumann, Shannon, and the rest of them4.3 right into the shade.

If we know something about the Universal set to which the
dictionary applies, we can sometimes guess a consistent mapping between
the ``real'' patterns of the subsets of that set and ``virtual''
patterns of the dictionary terms, possibly aided by visual cues such as
a ``picture of a tree'' next to its definition that help us establish at
least some provisional mappings. In essence, the dictionary represents
a code, and to break the code we have to determine a homologous
set of linkages between the dictionary and the system to which it
refers. Ultimately this task is made extraordinarily difficult because
there is no guarantee that any homology will be unique. Given the high
degree of degeneracy (redundancy) of human language it will almost
certainly not be unique4.4.

Dictionaries do not intrinsically specify a system of logic,
however, and a language is not simply the set of homologies
represented within the dictionary and some reference system.
Dictionaries (real ones, not idealized ones) are only rarely complete -
perhaps when they reference some ``simple'' closed system that is
capable of being well-defined (literally) such as the
``dictionary'' of a computer or mathematical language.

Because of their intrinsic incompleteness - a complete dictionary for
something like a real world would require the moral equivalent of a word
for every event in space-time that completely specifies the
homology between that event and all other events, plus the ability to
represent all higher order homologies built on top of the raw
physical homology - the ``language'' of human experience, of poetry, of
illogic and paradox and contradiction - a dictionary is most generally
an approximate, or coarse grained set of homologies, and
requires something more to aid in the abstraction of relationships before anything like a system of logic ensues.

We've encountered just the tip of this particular iceberg in our
discussions of sets, where ultimately the dictionary is what is
required to identify each object and sort it into its own identity
set when confronted with the Universal set. It is not enough to
identify an object as a ``tree'', it has to be able to identify an
object as this particular tree, with its own unitary and unique
existence, as of this particular moment in its existence. Where in fact the tree is made up of a dynamically changing set of molecules,
the molecules are made up of atoms, the atoms are made up of electrons
and nuclei, the nuclei are made up of protons and neutrons, and the
protons and neutrons are made up of quarks - ultimately a complete
definition of this tree extends to the subatomic level, to the
fundamental level, and extends through space and through time as a
set of intertwining relationships.

This in turn doesn't necessarily recognize or encode the relationships and structures that emerge at the higher degrees of
complexity. It is not at all easy to understand this tree's
particular role as a home for nesting birds and eventual source of
firewood based on an understanding, however complete, of its subatomic
structure4.5. Specifying relational operations is like
specifying the syntax of the language. We can define an apple quite
precisely (if we try hard enough) as a concatenation of specific
molecules that underwent a particular process of development in natural
history without ever mentioning that apples are good to eat, that an
apple a day keeps the doctor away, that a thrown apple can be used to
bean someone on the head, that deer are attracted to apple trees in the
back yard at certain times of year because they are good to eat
except those yards of healthy people who bean deer on the noggins with
apples any time they dare to show their furry little faces!

When we come to reason, we find that in addition to a set of definitions
(that are fundamentally arbitrary and certainly not ``obvious truths''
or ``provable'') we need to specify relationships in order to be able to
operate on the objects that are appropriately defined within the
theory. I leave this term deliberately vague - operating on an object
might (for example) be an action that ``transforms'' (in a sequential
reasoning sense, not a temporal sense) a defined object from one state
to another. Or it might be viewed as a sorting or categorization
operation, one that takes an object or subset from one set and places it
in another. Or it might assert a more abstract relationship between
objects or collective subsets that we discover we need as we proceed.
Ultimately such relationships function as rules of our system of
reason. There are two primary kinds of rules involved in formal logic.
One is the so-called rules of inference which are (as their name
suggests) a set of rules that permit one to ``infer'' provisional truth
relationships. The other is the set of axioms of the theory.

These two things are differentiated primarily by rules of inference
being presumably self-evident statements - in fact, the Laws of
Thought in disguise. They presumably ``come with the territory'' of set
theory, universes and mutually exclusive partitionings of identity
relationships, although I'm hoping that the previous chapters were
enough to make you a bit skeptical that this is in fact the case.
Axioms per se are simply unprovable assumptions, the hypotheses that
lead one to this system of reason (or this set theory, this branch of mathematics, this hypothesized universe, this computer's microarchitecture) and not that.

It is a fairly recent discovery that it is possible to choose different hypotheses and reason validly to different conclusions
even in that most precise and self-evidently obvious of mathematical
systems - ordinary geometry. It is worth repeating like a mantra that
while the sum of the angles in a triangle in plane geometry is
radians, in an infinity of other two-dimensional geometries
it is not. If we change the assumption that the two-dimensional
surface is ``flat'', the result goes away and is replaced by new,
different results.

I'm going to compress rules of inference into a very limited set
based on the set theory above, which does not require things like
the Law of Contradiction and the Law of Exclusion to universally operate
within the Universal set of the theory but rather to differentiate
that which is (in the Universal set, including the empty set) and that
which is not (is ). ``True'' and ``False'' will be particular sets
that may or may not be exclusive and exhaustive within the set Universe
distinct from ``Being'' and ``Non-Being''. The particular extension
that permits it to be applied to True and False categories will then
become an axiom of a particular system of reason.

This is a very good thing. It uses these two rules only to state the
truly obvious - ``Contradictions cannot occur'' - without specifying
precisely what a contradiction is. In fact, perhaps it is better
to think of it as being ``Contradictions do not occur''. The null
set (the impossible) is not in the Universal set, regardless of how
objects are parsed into nonempty and empty or true and false sets
withing the Universe.